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Calculus · Extrema

Critical points of x·(1 - x)

x \left(1 - x\right)

Step by step

  1. f(x) = x \left(1 - x\right)

    Critical points are where f′(x) = 0 or is undefined.

  2. f'(x) = 1 - 2 x

    Differentiate.

  3. x = \frac{1}{2}

    Solve f′(x) = 0.

  4. f''(x) = -2

    Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

  5. f''(\frac{1}{2}) = -2 \Rightarrow (\frac{1}{2}, \frac{1}{4}) \text{ is a local maximum}

Reveal the answer
(\frac{1}{2}, \frac{1}{4})\ \text{local maximum}